Chapter 4 Fuzzy Graph and Relation

Transcription

1 Chpter 4 Fuzzy Grph nd Reltion

2 Grph nd Fuzzy Grph! Grph n G = (V, E) n V : Set of verties(node or element) n E : Set of edges An edge is pir (x, y) of verties in V.! Fuzzy Grph ~ n ( ~ G = V, E) n V : set of verties n E : fuzzy set of edges etween verties

3 ! Exmple Fuzzy grph nd fuzzy reltion 0.8 M G Fig 4. Fuzzy grph

4 Fuzzy grph nd fuzzy reltion! Exmple n R + : nonnegtive rel numers. n n x R + nd y R + R = {(x, y) x y}, R R + R +. y x Fig 4. Fuzzy grph y x ( y loses to x)

5 Fuzzy grph nd fuzzy reltion! Exmple n The drkness of olor stnds for the strength of reltion in () n n n Reltion (, ) is stronger thn tht of reltion (, ). The orresponding fuzzy grph is shown in (). the strength of reltion is mrked y the thikness of line. () Fuzzy reltion R () Fuzzy grph Fig 4.3 Fuzzy reltion nd fuzzy grph

6 ! A grph nd fuzzy grph Fuzzy grph nd fuzzy reltion y y x - x - () Grph µ R (x, y) = x 2 + y 2 = - () Grph µ R (x, y) = x 2 + y 2 Fig 4.7 Fuzzy grph

7 α-ut of Fuzzy Grph! Exmple Appling α-ut opertion on fuzzy grph, for exmple A = {,, }, R A A is defined s follows. M R

8 α-ut of Fuzzy Grph 0.4 M R M R M R

9 ! Exmple µ R (x, y) = x/2 + y α-ut of Fuzzy Grph y y 0 2 x 0 2 x Fig 4.9 Grphil form of R Fig 4.0 Grphil representtion of R

10 α-ut of Fuzzy Grph! Exmple µ A (x) = x µ R (x,y) = x+y, x A, 0 y µ A (x) y 0 x 0 x Fig 4. Set µ A (x)= x Fig 4.2 Reltion µ A (x,y)= x+y, x A

11 α-ut of Fuzzy Grph! Exmple A={ x x lose to 2kπ, k = -,0,,2,.} µ A (x) = Mx[0, osx]. µ A (x) µ A (x) π π 2 π 3 π 0 3 π 2 π 3π 2 5π 3 2 π 7π 3 5π 2 x π π 2 π 3 π 0 3 π 2 π 3π 2 5π 3 2 π 7π 3 5π 2 x Fig 4.3 Set µ A (x)=osx 0 Fig 4.4 α-ut set A

12 Fuzzy Network! Pth with fuzzy edge V : risp set of nodes, R : reltion defined on the set V pth C i = (x i, x i2,..., x ir ), x ik V, k =, 2,..., r where (x ik, x ik+ ), µ R (x ik, x ik+ ) > 0, k =, 2,..., r- fuzzy vlue l for pth C i : the minimum possiility of onneting from x i to x ir. l (x i, x i2,..., x ir ) = µ R (x i, x i2 ) µ R (x i2, x i3 )... µ R (x ir-, x ir ) possile set of pths C(x i, x j ) = {(x i, x j ) (x i, x j ) = (x i = x i, x i2,..., x ir = x j )} vlue of mximum intensity pth l* (x i, x j ) = l (x i = x i, x i2,..., x ir = x j ) C(xi, xj) d

13 Fuzzy Network! Pth with fuzzy node nd fuzzy edge V : fuzzy set of nodes, R : fuzzy set of edge C i = (x i, x i2,..., x ir ), x ik V, k =, 2,..., r where, ( x ik, x ik+ ), µ R (x ik, x ik+ ) > 0, k =, 2,..., r- x ik, µ V (x ik ) > 0, k =, 2,..., r l(x i, x i2,..., x ir ) = µ R (x i, x i2 ) µ R (x i2, x i3 )... µ R (x ir-, x ir ) µ V (x i ) µ V (x i2 )... µ V (x ir ) (, 0.8) (, ) (, ) (d, 0.9) () Fuzzy network (node,edge)

14 Chrteristis of Fuzzy Reltion! Reflexive Reltion n For ll x A, if µ R (x, x) =! Exmple A = {2, 3, 4, 5} R : For x, y A, x is lose to y R n If x A, µ R (x, y), then the reltion is lled irreflexive. n If x A, µ R (x, y), then it is lled ntireflexive

15 Symmetri Reltion! Symmetri n (x, y) A A n µ R (x, y) = µ µ R (y, x) = µ! Antisymmetri n n (x, y) A A, x y µ R (x, y) µ R (y, x) or µ R (x, y) = µ R (y, x) = 0! symmetri or nonsymmeti n (x, y) A A, x y n µ R (x, y) µ R (y, x)! Perfet ntisymmetri n n (x, y) A A, x y µ R (x, y) > 0 µ R (y, x) = 0

16 Trnsitive Reltion! Definition n (x, y), (y, x), (x, z) A A n µ R (x, z) Mx [Min(µ R (x, y), µ R (y, z))]! If we use the symol for Mx nd for Min, the lst ondition e omes n µ R (x, z) [µ R (x, y) µ R (y, z)]! If the fuzzy reltion R is represented y fuzzy mtrix M R, we know t ht left side in the ove formul orresponds to M R nd right one t o M R2. Tht is, the right side is identil to the omposition of relti on R itself. So the previous ondition eomes, n M R M R2 or R R 2

17 Trnsitive Reltion! Trnsitive reltion exmple For (, ), we hve µ R (, ) µ R2 (, ) For (, ), µ R (, ) µ R2 (, ) We see M R M R2 or R R Fig 4.20 Fuzzy reltion (trnsitive reltion)

18 Clssifition of Fuzzy Reltion! Fuzzy Equivlene Reltion Definition(Fuzzy equivlene reltion) () Reflexive reltion x A µ R (x, x) = (2) Symmetri reltion (x, y) A A, µ R (x, y) = µ µ R (y, x) = µ (3) Trnsitive reltion (x, y), (y, z), (x, z) A A µ R (x, z) Mx[Min[µ R (x, y), µ R (y, z)]] y

19 Clssifition of Fuzzy Reltion! Exmple (Grph of fuzzy equivlene reltion ) d d d

20 Clssifition of Fuzzy Reltion! Applition : Prtition of sets set A is done prtition into susets A, A 2,... y t he equivlene reltion! Exmple d e d.0 e.0 A A A 2 d e

21 Clssifition of Fuzzy Reltion! Applition 2 : Prtition y α-ut n α-ut equivlene reltion R α µ R (x, y) = if µ R (x, y) α, x, y A i = 0 otherwise

22 Clssifition of Fuzzy Reltion! Exmple π(a/r )= {{, }, {d}, {, e, f}} d e f d e.0.0 f.0 α = α = 0.4 α = α = 0.8 α =.0 d e f d e f d e f d e d e f f

23 Clssifition of Fuzzy Reltion! Fuzzy Order Reltion Definition(Fuzzy order reltion) () Reflexive reltion x A µ R (x, x ) = (2) Antisymmetri reltion (x, y) A A µ R (x, y) µ R (y, x) or µ R (x, y) = µ R (y, x) = 0 (3) Trnsitive reltion (x, y), (y, z), (x, z) A A µ R (x, z) Mx[Min(µ R (x, y), µ R (y, z))] y

24 Clssifition of Fuzzy Reltion! Exmple (fuzzy order reltion ) d

25 Clssifition of Fuzzy Reltion! Definition(Corresponding risp order) i) if µ R (x, y) µ R (y, x) then µ R µ R ii) if µ R (x, y) = µ R (y, x) then µ R ( x, y) = µ ( y, x) = R 0 ( x, y) = ( y, x) = 0

26 Clssifition of Fuzzy Reltion! Exmple 0 d d d Crisp order reltion otined from fuzzy order reltion)

27 Clssifition of Fuzzy Reltion! Definition(Dominting nd dominted lss) R (x, y) > 0, Sy tht x domintes y nd denote x y. ) The one is dominting lss of element x. Dominting lss R [x] whih domintes x is defined s, µ R [x] (y) = µ R (y, x) 2) The other is dominted lss. Dominted lss R [x] with ele ments dominted y x is defined s, µ R [x] (y) = µ R (x, y)

28 Clssifition of Fuzzy Reltion28! Exmple n n Dominting lss of element nd R [] = {(,.0), (, 0.7), (d,.0)} R [] = {(,.0), (d, 0.9)} dominted lss y R [] = {(,.0), (, )} d d d

29 Clssifition of Fuzzy Reltion! fuzzy upper ound of suset = {x, y} R [ x] x A' A'! Exmple n fuzzy upper ound A' = {, } R [] R [] = {(,.0), (, 0.7), (d,.0)} {(,.0), (d, 0.9) } = {(, 0.7), (d, 0.9)}

30 Fuzzy Morphism! Homomorphism R A A, S B B homomorphism funtion h : A B from (A, R) to (B, S) For x, x 2 A (x, x 2 ) R (h(x ), h(x 2 )) S If two elements x nd x 2 re relted y R, their imges h(x ) nd h(x 2 ) re lso relted y S

31 Fuzzy Morphism! Strong homomorphism R A A, S B B h : A B For ll x, x 2 A, (x, x 2 ) R (h(x ), h(x 2 )) S For ll y, y 2 B, if x h - (y ), x 2 h - (y 2 ) then (y, y 2 ) S (x, x 2 ) R

32 Fuzzy Morphism! Fuzzy homomorphism Fuzzy reltion R A A, S B B Funtion h : A B stisfies For ll x, x 2 A µ R (x, x 2 ) µ S [h(x ), h(x 2 )] The strength of the reltion S for (h(x ), h(x 2 )) is stronger thn or equl to the tht of R for (x, x 2 ).

33 Fuzzy Morphism! Fuzzy strong homomorphism Fuzzy reltion R A A, S B B funtion h : A B stisfies For ll x j A j, x k A k, A j, A k A y =h(x j ), y 2 = h(x k ) y, y 2 B, (y, y 2 ) S, Mx µ R (x j, x k ) = µ S (y, y 2 ) xj, xk

34 Exmples of Fuzzy Morphism! Exmple R d 0.6 S α β γ 0.8 α β d 0.6 γ 0.6 n ll (x, x 2 ) R A hs the reltion (h(x ), h(x 2 )) S in B n µ R (x, x 2 ) µ S (h(x ), h(x 2 )) n h() = β, h(d) = γ, µ R (, d) = 0 µ S (β,γ) = 0.6 h :, α β d γ

35 Exmples of Fuzzy Morphism! Exmple 0.6 α β d 0.6 γ h : A B

36 Exmples of Fuzzy Morphism! Exmples of Fuzzy Strong Morphism R d e S α β γ.0 α.0 β.0.0 d 0.9 γ e.0 n (β,γ) S, µ S (β,γ) =, n h - (β) = {, }, h - (γ) = {d, e} h : α, β d, e γ n Mx [µ R (, d), µ R (, e)] = Mx [, ] = = µ S (β,γ)

37 Exmples of Fuzzy Morphism! Exmple α β d e γ R A A h : A B S B B